mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, a

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...

is antisymmetric on (or with respect to) an index subset if it alternates

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or ...

(+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ''covariant'' or all ''contravariant''. For example,

T_ = -T_ = T_ = -T_ = T_ = -T_

holds when the tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of ''each'' pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant

tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

k

may be referred to as a differential

k

-form, and a completely antisymmetric contravariant tensor field may be referred to as a

k

-vector field.

Antisymmetric and symmetric tensors

A tensor A that is antisymmetric on indices

i

and

j

has the property that the

contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...

with a tensor B that is symmetric on indices

i

and

j

is identically 0. For a general tensor U with components

U_

and a pair of indices

i

and

j,

U has symmetric and antisymmetric parts defined as: : Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in

U_ = U_ + U_.

Notation

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor M,

M_ = \frac(M_ - M_),

and for an order 3 covariant tensor T,

T_ = \frac(T_-T_+T_-T_+T_-T_).

In any 2 and 3 dimensions, these can be written as

T_ &= \frac \, \delta_^ T_ . \end

where

\delta_^

is the generalized

Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...

, and we use the

Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...

to summation over like indices. More generally, irrespective of the number of dimensions, antisymmetrization over

p

indices may be expressed as

T_ = \frac \delta_^ T_.

In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:

T_ = \frac(T_ + T_) + \frac(T_ - T_).

This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.

Examples

Totally antisymmetric tensors include: * Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric). * The

electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. ...

F_

electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...

. * The Riemannian volume form on a

pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

Antisymmetric and symmetric tensors

Notation

Examples

See also

Notes

References

External links